Imagine a hypothetical planet “flat earth” of thickness ‘t’ having same gravity on its surface as “our Earth”. Assume “Flat earth” to be infinite plate of same uniform density as “our Earth”, which is a uniform solid sphere of Radius R. if $t = \frac{4 R}{n}$ , find the value of $n ?$

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answer is 6.

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Detailed Solution

Gauss law in gravitation

$E = \frac{GM}{R^{2}}$ on the earth surface in the form of sphere [our Earth]

$\int \vec{E} \cdot \vec{ds} = - 4 π G [\min]$

$\Rightarrow 2 \int E ds \cos 180 = - 4 πG [m_{in}]$

$\Rightarrow 2 \int EA = 4 πG [m_{in}] \Rightarrow 2 EA = 4 πG [ρ A t]$

$\Rightarrow \frac{2 GM}{R^{2}} \cdot A = 4 πGρ At$

$\Rightarrow \frac{2 G \frac{4}{3} {πR}^{3} ρ A}{R^{2}} = 4 π G ρ A t$

$\Rightarrow t = \frac{2 R}{3} = \frac{4 R}{6}$ $∴ n = 6$

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